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Blowup and Dissipation in a Critical-Case Unstable Thin film Equation ∗ T

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Blowup and Dissipation in a Critical-Case Unstable Thin film Equation ∗ T Euro. Jnl of Applied Mathematics (2004), vol. 15, pp. 223–256. c 2004 Cambridge University Press 223 DOI: 10.1017/S0956792504005418 Printed in the United Kingdom Blowup and dissipation in a critical-case unstable thin film equation ∗ T. P. WITELSKI1,A.J.BERNOFF2 and A. L. BERTOZZI3 1Department of Mathematics and Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708-0320, USA email: [email protected] 2Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA email: [email protected] 3Departments of Mathematics and Physics, Duke University, Durham, NC 27708-0320, USA email: [email protected] (Received 4 April 2003; revised 1 December 2003) We study the dynamics of dissipation and blow-up in a critical-case unstable thin film equation. The governing equation is a nonlinear fourth-order degenerate parabolic PDE derived from a generalized model for lubrication flows of thin viscous fluid layers on solid surfaces. There is a critical mass for blow-up and a rich set of dynamics including families of similarity solutions for finite-time blow-up and infinite-time spreading. The structure and stability of the steady-states and the compactly-supported similarity solutions is studied. 1 Introduction This paper studies the dynamics of the fourth-order nonlinear parabolic partial differential equation ∂h ∂ ∂h ∂ ∂3h = − h3 − h (1.1) ∂t ∂x ∂x ∂x ∂x3 for non-negative finite-mass initial data on a periodic domain, −1 6 x 6 1. We show that this problem has a rich structure including equilibrium solutions and continuous families of similarity solutions for both finite-time blow-up and self-similar spreading. There is a vast body of literature on similarity solutions and the formation of singularities in partial differential equations – for example see the references in [2, 3, 13, 30, 63]. Classic studies considered the dynamics of blow-up resulting from interactions between nonlinear terms and second-order spatial operators, as in the nonlinear Schrodinger equation [21, 28, 46, 48, 55, 60] and in semilinear heat equations [2, 20, 29, 30, 31, 32, 33, 34, 47, 57]. The blow-up dynamics in (1.1) are governed by the the interaction between nonlinear second- and fourth-order terms, and as such it represents a higher-order analogue of these second-order model problems. The dynamics of (1.1) share many common features with the previous models, such as the existence of multi-bump similarity solutions [21, 22, 55, 56], ∗ Current address: Mathematics Department, UCLA, Box 951555, Los Angeles, CA 90095-1555, USA. Email: [email protected] 224 T. P. Witelski et al. but the solutions of this nonlinear degenerate problem are weak compactly-supported ‘droplet’ solutions. Equation (1.1) is a special case of the longwave-unstable generalized thin film equation, − m − n ht = (h hx)x (h hxxx)x, (1.2) where h(x, t) gives the height of the evolving free-surface. The exponents m, n correspond to the powers in the destabilizing second-order and the stabilizing fourth-order diffusive terms, respectively. This class of model equation occurs in connection with many physical systems involving fluid interfaces [50, 53]; when n =1andm = 1 it describes a thin jet in a Hele–Shaw cell [1, 13, 23, 25, 36]. When n =3andm = −1 it describes van der Waals driven rupture of thin films [62, 65, 66, 67], and for n = m = 3 it describes fluid droplets hanging from a ceiling [26]. For n =0andm = 1, equation (1.2) is a modified Kuramoto–Sivashinsky equation which describes solidification of a hyper-cooled melt [8, 17]. Over the past 15 years, these models have also been the focus of rigorous and extensive mathematical analysis [4, 5, 12, 15, 18, 42, 43, 44, 45, 54]. In this paper, we focus on equation (1.2) with n =1andm = 3 because of its special role as a critical case between problems where solutions blow-up (see Figure 1) and problems whose solutions are bounded for all times. Questions on blow-up in (1.2) were raised by Hocherman & Rosenau [39]. They proposed the existence of critical exponents that would separate those equations that have blow-up, h →∞, from those that do not. That is, if the strength of the destabilizing term is sufficient to overcome the regularizing influence of the fourth-order term, then blow-up can take place. The rigorous analysis of the blow-up problem was recently studied in two papers by Bertozzi & Pugh [16, 17]. They proved [16] that blow-up is impossible for m<n+ 2, disproving the original conjecture of [39] that m = n is critical. In the second paper [17], the special case n = 1 is studied on the line for compactly supported initial data. They prove the existence of a solution that blows up in fi- nite time for m > 3=n +2.Consequently,forn =1,m = n + 2 = 3 is the critical exponent separating equations with possible finite-time blow-up from problems where the solutions are always bounded. As a critical case equation, (1.1) has many interesting characteristics that we will explore: a finite critical mass for blow-up, two classes of continuous families of self-similar solutions with compact support, and delicate interactions that can occur between pinch-off and blow-up. Our results build on a single framework that will serve to unify the dynamics of (1.1) in three regimes: (i) near equilibrium, (ii) approaching finite- time blow-up, and (iii) infinite-time diffusive spreading. We will explore the connections between these different classes of solutions. For much of our work, it is convenient to write (1.1) in a slightly different form, as ∂h ∂ ∂p ∂2h + h =0,p≡ 1 h3 + , (1.3) ∂t ∂x ∂x 3 ∂x2 where p defines a pressure function. This form stems from the interpretation of (1.1) as a generalization of the Reynolds lubrication equation for thin films of viscous fluids [53]. In this context, the terms in the pressure describe a body force on a thin fluid layer due to the cube of the thickness of the layer, and a surface tension contribution given by the linearized curvature of the free-surface of the layer, respectively. The pressure is a Dynamics of a critical-case thin film equation 225 Figure 1. Finite-time blow-up for a solution of PDE (1.1) in a periodic domain. key part of the analysis of the similarity solutions. We study initial value problems for (1.1) on the interval −1 6 x 6 1 with periodic boundary conditions, h(x +1)=h(x − 1), and non-negative initial data. Results from this problem can be related to the Neumann boundary value problem and the short-time dynamics of the Cauchy initial problem with compact initial data, under appropriate conditions [42, 43]. The total mass of the solution is given by 1 M = hdx; (1.4) −1 the mass is conserved for all times. We shall use the mass as a control parameter to distinguish classes of initial data that will lead to different dynamics. Another fundamental global property of solutions of (1.1) is the monotone dissipation of the energy functional, 1 E 1 2 − 1 4 = 2 hx 12 h dx, (1.5) −1 at the rate E 1 d 2 = − h (∂xp) dx 6 0. (1.6) dt −1 It is possible to make use of the energy to describe: (i) the stability and dynamics of the solution [44]; and (ii) the evolution as a gradient flow in an appropriate weighted H−1 norm [9]. The remainder of this article is as follows. In § 2, we review results on blow-up of solutions with negative energy (1.5). We identify a critical mass Mc ≡ 2π 2/3below, which the solutions to (1.1) remain bounded for all time for both the periodic and Cauchy problems. In § 3, we use dimensional analysis to obtain the scalings for the first-type similarity solutions of (1.1). The same set of similarity variables provide a framework for studying both classes of self-similar solutions: finite-time blow-up and infinite-time spreading. In § 4, we examine the structure of these two classes of similarity solutions and the steady states of (1.1). These solutions all exist as equilibria of an equation we 226 T. P. Witelski et al. call the similarity PDE, which is a generalization of (1.3). The connections between these states are explored within this framework, somewhat analogously to the study of blow-up solutions and steady states [19]. In § 5, the linear stability of these equilibria is analyzed. For the two classes of similarity solutions, the influence of the symmetries of the PDE must be considered in studying the spectrum [9, 63, 66]. Finally, in § 6, further issues, problems, and open questions for the nonlinear dynamics of (1.1) are addressed using numerical simulations. 2 Conditions on finite-time blow-up Perhaps the most dramatic behavior exhibited by solutions of (1.1) is that of finite-time blow-up, see Figure 1. The occurrence of blow-up can be noted from an argument based on the evolution of the second moment of the solution, as suggested by Bernoff [17], d x2hdx = − 1 h4 dx +3 h2 dx =6E. (2.1) dt 2 x Since the energy (1.5) is monotone decreasing, (1.6), we have a bound on the evolution of the second moment in terms of the initial energy, E0, d x2hdx 6 6E . (2.2) dt 0 If the initial energy is negative, then the second moment will become negative at a finite time.
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